Abstract:
We present the first nontrivial space­time tradeoff lower bounds for hyperplane and
halfspace emptiness range queries. Our lower bounds apply to a general class of geometric
range query data structures called partition graphs. Informally, a partition
graph is a directed acyclic graph that describes a recursive decomposition of space. We
show that any partition graph that supports hyperplane emptiness queries implicitly
defines a halfspace range searching data structure in the Fredman/Yao semigroup arithmetic
model with the same time and space bounds. Thus, results of Brönnimann, Chazelle,
and Pach imply that any partition graph of size s that supports hyperplane
emptiness queries in time t must satisfy the inequality
std =
Omega
((n / log n)d-(d-1)/(d+1)).
Using different techniques, we improve previous lower bounds for
Hopcroft's problem -- Given a set of points and hyperplanes, does any hyperplane
contain a point? -- in dimensions four and higher. Using this offline result, we show that
for online hyperplane emptiness queries,
Omega(nd/polylog n)
space and preprocessing time is required to acheive polylogarithmic query time, and that
Omega(n1-1/d polylog n)
query time is required if only O(n polylog n) space is available.
These two lower bounds are optimal up to polylogarithmic factors. For two­dimensional
queries, we obtain an optimal continuous tradeoff between these two extremes. Finally,
using a lifting argument, we show that the same lower bounds hold for offline and online
halfspace emptiness queries in Rd(d+3)/2.

Update (26 Sep 2000):Braß and
Knauer, using recent work of
Bárány, Harcos,
Pach, and Tardos, recently discovered a set
of n points and m hyperplanes, where at most d hyperplanes pass
through any point, with a large number of point-hyperplane incidences. Their construction
immediately improves several of my lower bounds in d>2 dimensions.

For Hopcroft's problem, we have the new lower bound
Omega(n log m + nd/(2d-1)m(2d-2)/(2d-1) +
n(2d-2)/(2d-1)md/(2d-1) + m log n).
This is an improvement for all n1/d << m
<< nd.

For online hyperplane emptiness queries, we have new space-time tradeoffs
st2d-2 = Omega(nd) and
std/(d+1) = Omega(n2), and the
analagous preprocessing-query tradeoffs. This is an improvement for all n
<< s << nd.

There are similar improvements for offline and online halfspace emptiness queries in
dimensions nine and higher.

All these improvements follow directly from the techniques in my paper.